Blog #10: WS4.1, #1

1. Mass bends space-time! This is a prediction of general relativity, but fortunately we can heuristically derive the effect (up to a factor of 2) using Newtonian mechanics and some simplifying assumptions. Consider a photon of "mass" \(m_{\gamma}\) passing near an object of mass \(M_L\); we'll call this object a "lens" (the 'L' stands for 'lens', which is the object doing the bending). The closest approach (b) of the photon is known as the impact parameter. We can imagine that the photon feels a gravitational acceleration from this lens which we imagine is vertical. 

a) Give an expression for the gravitational acceleration in the vertical direction in terms of M, b, and G. 

To do this, we can consider two different ways of calculating force and set them equal to each other, then solve for the acceleration. 

\(F = m_{\gamma}a = \frac{GM_Lm_{\gamma}}{b^2}\) 

\(a = \frac{GM_L}{b^2}\)

b) Consider the time of interaction, ∆t. Assume that most of the influence the photon feels occurs in a horizontal distance 2b. Express ∆t in terms of b and the speed of the photon. 

\(v_{photon} = c\) 

\(c = \frac{2b}{∆t}\) (speed = displacement/time)

\(∆t = \frac{2b}{c}\)

c) Solve for the change in velocity, ∆v, in the direction perpendicular to the original photon path over this time of interaction.

We can assume that the photon's gravitational acceleration is constant as it moves past the lens. 

\(∆v = ∆ta = \frac{2b}{c}\frac{GM_L}{b^2} = \frac{2GM_L}{cb}\) 

d) Solve for the deflection angle (\(\alpha\)) in terms of G, \(M_L\), b, and c using the answers from parts a, b, and c. 

\(tan(\alpha) = \frac{∆v}{c}\)

\(tan(\alpha) = \frac{2GM_L}{c^2b}\) 

Because of the small angle approximation, \(tan(\alpha) = \alpha\) 

\(\alpha = \frac{2GM_L}{c^2b}\) 

This result is a factor of 2 smaller than the correct, relativistic result, so \(\alpha = \frac{4GM_L}{c^2b}\)